Climate Modeling using a Leaky Integrator

[VillagePlank]

With reference to the time lag on natural processes, and with small signals getting 'lost' I've been a good doggy, and I've generalised ......

I see it more as a leaky integrator system (for each component) of the weather:

Where alpha and beta are positive constants, a is the current state, and s is the rate of change to a.

This system is called a leaky integrator because it is analagous to a bucket of water where the bucket has a hole in it, and there is a hose filling up the bucket with water. To clarify .... the rate at which the hose is filling up the bucket is s, and the rate at which water is leaking out of the bucket is directly proportional to a - the depth of water in the bucket. The constants change the system for various rates of leakage etc etc.

As always, a picture says a thousand words ... here's an excel graph modelling a hypothetical system ....

Here, s, is added to the system for a short period of time, and then withdraw to zero. at t=16. You can see clearly, given the constants, that although the system reacts quickly to input, it reacts very slowly by itself when input is withdrawn.

So, let's have a go at modelling the 11 year sun cycle using the same constants. As expected it settles down into some sort of equilibrium, which is, that all other things are held constant, the system settles down into a predictable state - all other things remaining constant have been added as an additional factor of 0.05 added to s, and denoted as c.

Now, let's try increment the constant,c, by a very small amount, say 0.01, from about t=50, for each t.

Here, you can see that the solar cycle is still evident in a rising a; has the very small increments of c modified the picture somewhat? It's not evident from the sinusoidal curves of the solar effect is it? But taken on the long term which way does the trend go?

Of course, this is just linear additions of c, when, I suspect, that CO2 are more exponential, and also, the model reacts well to reaching a higher equilibrium if you turn off c at some arbitrary late stage.

But.... the point is - at which point did you notice the solar cycles disappear? The answer is, of course, they simply didn't. Not only that but the solar cycles are still, by a factor of three, more influential on the outcome of a, than c - even though the very small value of c (1/3rd of solar) has pushed up a to almost double.

EDIT: the more observant of you will notice that I used da/dt, not dy/dt - and that is because a stands for activation of a membrane on a neural network (and I couldn't be ar5ed to change the equation)

Can I have a doggy biscuit, now, please?

Edited December 16, 2008 by VillagePlank

[VillagePlank]

... and just for a laugh, here's what happens if we have a nuclear war, kill off the sunlight, and c emmissions reduce to 0.25.

Only one thing for it, then: where's that bunker - at the end of the flat earth, perhaps? Or, more likely, I think, to be strapped on one of those turtles, because, of course, it's turtles all the way down ....

[Gray-Wolf]

and what if 's' was able , over time ,to seal the puncture in the bucket?

[VillagePlank]

So that a couldn't reduce?

Well it'd heat up an awful lot, then.

[Admiral_Bobski]

Here's a point in a post above that I think needs addressing:

I apologise for that comment - I was exaggerating your claim (unfairly, perhaps) to make a point. Why would you expect temperatures to be "quite a bit lower" than they are now? And how much is "quite a bit"? Presumably your expectations are once again driven by the preconceived idea that solar effects are minimal, and that the lag is negligible (2 years or less).

The reason is that if solar effects were large, we would surely expect there to be a large cooling as a result of solar activity changes. In reality, we haven't seen any significant cooling (maybe 0.1-0.2C at the very most) and much of this can be explained by ENSO anyway, so the suggestion is that either recent solar activity has had a minimal effect, or something else must be offsetting the large effect that it has had.

Hi TWS,

I think you may have missed the gist of my argument, which is that Earth climate reaction to solar changes is subject to a time-lag (and, further, the suggestion that around fifty years of over-active Sun may have had a cumulative effect on Earth's climate, as I think VP was analogising).

If Earth's climate lags behind solar activity then we wouldn't expect the current solar minimum to have affected Earth climate yet. I do not see that this is a particularly off-the-wall suggestion, since several scientific studies allude to a climatological time-lag - it's the length of the lag that is at issue, I think.

CB

[VillagePlank]

Just in case anyone thinks I making my last posts up .... here's it more formally, and, using my modified version (I use the beta constant version added as recommended in neural network stuff) here's the chart, with s increasing by 1, per t, and the constants set as displayed.

Cap'm: I am not really analagising (sp?) anything, really, just pointing out two critical points:

(i) The CO2 signal is much much smaller than any natural signal - but it is still there, and is easily seen. EDIT: a correction, there is small but cumulative temperature increase signal in the temperature record (trying to be AGW-Agnostic, here!)

(ii) Time lags aren't obvious even with a linear differential system such as the leaky integrator - which can generate all sorts of strange shapes (including sawtooth patterns) given the right set of constants

I'm not saying I'm right, and I'm not saying either +AGW or -AGW are right, I'm just saying it is mathematically possible, and models that model the real world, such as the leaky integrator - which models potential on a neuron's membrane - already use this, and no one seems to mind time lags, or small insignificant signals (vitally important in neural nets - it's what makes them resilient to error)

Edited December 16, 2008 by VillagePlank

[Admiral_Bobski]

Fair points well made, VP! I read it as a mathematical analogy, but I suppose it's not really an analogy so much as a simplified mathematical model. Good work though.

Since you've got the model all worked out there, if you've got time do you think you could mock up an example with the 11-year cycle varying randomly in intensity, so we can see how the trend alters over time? You could do it twice - with and without c.

Cheers

CB

[Devonian]

With reference to the time lag on natural processes, and with small signals getting 'lost' I've been a good doggy, and I've generalised ......

I see it more as a leaky integrator system (for each component) of the weather:

.....

Can I have a doggy biscuit, now, please?

Please, Sir, can you explain that again? . Or, god I hate maths 'let x = 7 and Y be an inverse proportional, then dx/dt = 'heck i don't know '' . Climate I get in one read, maths I read it again and again and nothing drops . Or, I kind of get it but it's like that branch just out of reach, I can see it but I just can't grab it somehow . It might be because I'm not quite sure how the text relates to the graphs (labels?). This "But.... the point is - at which point did you notice the solar cycles disappear? The answer is, of course, they simply didn't. Not only that but the solar cycles are still, by a factor of three, more influential on the outcome of a, than c - even though the very small value of c (1/3rd of solar) has pushed up a to almost double." I don't get. Which graph? The third one? Then the line wiggles due to solar but rises due to CO2? Or the wiggles are three times as big as the rate of rise?

Or, are you trying to explain how an input over time works and what happens if you change any input?

[VillagePlank]

I presume you get chart one? Given an input of s (the bottom line) the leaky integrator gives the curve, denoted by a. This shows that you can switch off the input, s, and it takes ages to reduce in value.

Chart two shows what happens if you vary 's' rather like solar cycles - look at the blue line at the bottom, and compare to the red line at top. Because the system was started at zero it takes a little while to reach equilibrium. That is, using the bucket metaphor, there is, eventually, enough quantity of water in the bucket to, more or less, force out the same amount of water that is coming in.

Chart three shows what happens if you upset the input, in this case, the same as chart two, by increasing the input by a very small cumulative amount. It upsets the system so much that it never reaches equilibrium - at least not in this time frame. It alos shows how a very very small value can have very profound effects on a linear system (we all know it's affect on non-linear, or chaotic, systems)

Does this help?

[Devonian]

Does this help?

I'm getting there, slowly :lol: ....

[VillagePlank]

I'm getting there, slowly ....

I've sent you the spreadsheet - let me know how you get on

[VillagePlank]

Since you've got the model all worked out there, if you've got time do you think you could mock up an example with the 11-year cycle varying randomly in intensity, so we can see how the trend alters over time? You could do it twice - with and without c.

Bit busy, today, so I've done a basic one.

The following chart uses sunspot count as it's input, s, and the constants are displayed on the chart, and it uses the leaky integrator (discussed above) to give the quantity of 'a' a 'memory' I make no theory associated with this!

EDIT: sunspot numbers are from here

EDIT2: Does anyone know where I can find the data for volcanic eruptions and magnitude of volcanic eruptions since 1700AD or similar, please?

Edited December 17, 2008 by VillagePlank

[Admiral_Bobski]

That's fantastic VP - exactly what I was looking for! Thank you very much for taking the time to do that for me

Now, I'm assuming that alpha and beta are arbitrary values, so this isn't necessarily an accurate picture of the way the change in temperature is affected by the change in solar activity in the real world, but it illustrates my basic point.

Look at "s", the blue line (solar activity) circa 1956 - this is the highest peak, and yet the highest peak in "a", the temperature proxy in this argument, isn't until around 1995. By dint of our leaky integrator we find that the more recent cycles, while lower, have caused "a" to peak far later than the peak in solar activity due to the time it takes for the extra input in 1956 to "drain" out of the system.

Does anyone agree with this basic argument?

CB

[VillagePlank]

I'm assuming that alpha and beta are arbitrary values, so this isn't necessarily an accurate picture of the way the change in temperature is affected by the change in solar activity in the real world, but it illustrates my basic point.

Look at "s", the blue line (solar activity) circa 1956 - this is the highest peak, and yet the highest peak in "a", the temperature proxy in this argument, isn't until around 1995. By dint of our leaky integrator we find that the more recent cycles, while lower, have caused "a" to peak far later than the peak in solar activity due to the time it takes for the extra input in 1956 to "drain" out of the system.

Alpha and Beta are arbitrary values, but the description of how it takes time to, firstly, reach a peak, and to secondly, drain from a peak, is entirely the point of a leaky integrator. Everyone (I think) in climatology talks about how the oceans have memory and this model shows what happens when you give energy input a 'memory'

Again, it's worth saying, that this is only a model of 'memory', where, apart from the sunspot values, everything else has been set to an arbitrary value to make it easy to spot the 'memory': an important and crucial caveat.

Edited December 17, 2008 by VillagePlank

[VillagePlank]

... and this is simply mischievous ....

[Admiral_Bobski]

Alpha and Beta are arbitrary values, but the description of how it takes time to, firstly, reach a peak, and to secondly, drain from a peak, is entirely the point of a leaky integrator. Everyone (I think) in climatology talks about how the oceans have memory and this model shows what happens when you give energy input a 'memory'

Again, it's worth saying, that this is only a model of 'memory', where, apart from the sunspot values, everything else has been set to an arbitrary value to make it easy to spot the 'memory': an important and crucial caveat.

Duly noted, VP - that's what I thought. So basically if we knew what the real world values of alpha and beta were, plus the value (static or varying) of "c" (or rather c,d,e,f...) then we might be able to cobble together a more accurate model.

As you say, though, your graph shows the principle of my argument perfectly: namely that a lack of increase in solar activity over the last 20-30 years does not necessarily negate the possibility that the Sun could be largely responsible for observed warming trends over that period.

If there is, indeed, a solar activity/climate time lag then the big cycle of the 50s, in addition to higher-than-average cycles since then, could legitimately have caused warming by a cumulative effect.

Thanks again for taking the time to plot that graph for us, VP

CB

... and this is simply mischievous ....

Nothing wrong with a bit of mischief at Christmas time, is there?

Edited December 17, 2008 by Captain_Bobski

[VillagePlank]

Nothing wrong with a bit of mischief at Christmas time, is there?

Just remember that you saw it here, first

So basically if we knew what the real world values of alpha and beta were, plus the value (static or varying) of "c" (or rather c,d,e,f...) then we might be able to cobble together a more accurate model.

Yeah - I've been thinking about that .... working out the Milankovich cycles as I type, where, I presume, they will form a multiple of 's' (being that the closer to the sun, the more effect sunspots have? - I might just use sun/earth distance, actually - being that I'm far too lazy.....)

Complete conjecture, of course

[Admiral_Bobski]

Yeah - I've been thinking about that .... working out the Milankovich cycles as I type, where, I presume, they will form a multiple of 's' (being that the closer to the sun, the more effect sunspots have? - I might just use sun/earth distance, actually - being that I'm far too lazy.....)

Complete conjecture, of course

Hi VP,

that's a good place to start. I wonder also if there's any way of adding negating factors like major volcanic eruptions at the appropriate intervals (Vesuvius, Mount St Helens etc)? It would certainly be interesting to see how adding complications to the system alters the effect on the temperature proxy, "a".

Great work!

CB

[VillagePlank]

Yeah, it's simply a subtraction from s, so that, rearranged ...

s = eb-d

where e=sunspot count, b=deviation of earth sun distance from AU, and d=volcanic disruption, and then you plug in s to the leaky integrator as before.

Also, I think you'd need to adjust s for the effect of different land masses. For instance, we might assume that the model stands for oceans (known time lag?) but for the other 1/3 you might want to reduce 'memory' of land, and eliminate memory from ice. Perhaps. I don't know.

The difficulty in all these parameters will be quantification (how much did that volcano stop incoming sunspot energy, say) and scaling (how can we scale our value to fit the model)

I think I've got the scaling, and quantification right for the earth/sun distance by using it as a multiplier of the difference to AU (the average earth sun distance) so that if the sun is further away, the mutliplier might by 0.95 (reducing s) and if it's closer it might be 1.03 (increasing s)

[Admiral_Bobski]

Blimey - it's starting to get surprisingly complicated already, isn't it? (But that's a whole different argument ) You're right about quantification - I mean, we're already stumbling since we can't quantify the basic activity/climate lag (somewhere between 0 and 100 years has been mentioned, though it might be fairer to say there's a lag of 0-10 years in some parts of the system and a lag of up to 100 years in other parts of the system - complicated either way).

It might be a good place to start by assuming a basic time lag (as you have) and then quantifying other factors as fractions of the initial assumption.

Since we're just (at this point!) going for a rough analogy of the climate system, should we assume that all major volcanic eruptions block out 50% of incoming solar radiation for a period of, say, 2 years? Just to see what happens to "a".

Shall we complicate things as simplistically as possible (if that makes sense)?!

Is it also worth making changes one at a time to the equation and running a separate graph for each change? That way we can see, graphically, what a difference each factor has made to the previous run.

CB

[VillagePlank]

Since we're just (at this point!) going for a rough analogy of the climate system, should we assume that all major volcanic eruptions block out 50% of incoming solar radiation for a period of, say, 2 years? Just to see what happens to "a".

Do you have list of 'major' eruptions since 1700AD? (and their year/month)

Edited December 17, 2008 by VillagePlank

[Admiral_Bobski]

Do you have list of 'major' eruptions since 1700AD? (and their year/month)

Thanks to the wonderful Encarta, here's a list of every eruption that has been classed as "Major" since 1700:

Papandayan, Java - 1772

Laki, Iceland - 1783

Unzen, Japan - 1792

Tambora, Indonesia - 1815

Krakatau, Indonesia - 1883

Santa Maria, Guatemala - 1902

Pelee, Martinique - 1902

Taal, Philippines - 1911

Kelut, Java - 1919

Merapi, Indonesia - 1930

Rabaul, P. New Guinea - 1937

Lamington, PN Guinea - 1951

Hibok Hibok, Philippines - 1951

Agung, Indonesia - 1963

Mt St Helens, USA - 1980

El Chichon, Mexico - 1982

Nevado del Ruiz, Colombia - 1985

Lake Nyos, Cameroon - 1986

Pinatubo, Philippines - 1991-1996

Unzen, Japan - 1991

Mayon, Philippines - 1993

That's a bigger list than I was expecting! Is it worth assuming the same effect for all volcanoes? I presume there are many subtle levels of "Major", but it'll take a bit of digging to find out how Major each eruption was. Perhaps we should reduce their influence to 20% or 25%, so that their effect doesn't overwhelm the background data. It might also be worthwhile reducing the duration of their effect from 2 years to just one year and see what happens.

I leave it in your more-than-capable hands!

B)

CB

PS - Unfortunately the Encarta page doesn't give months of each eruption. If we assume that they all took place in June...? Also, for the years with multiple eruptions (1902, 1951, 1991) we should add the effects, I guess, so two eruptions would, assuming 20% per volcano, block 40% of incoming solar radiation (or would the second eruption only block 20% or the remaining 80%, in which case the total would be 36% blockage...?)

As for Pinatubo, which is listed as occurring over a span of five years, what do you think best? 20% every year for 5 years, or a gradual rise and fall from, say, 10% up to 20% and back down again?

EDIT - I've just noticed that the list only goes up to 1993 - additionally there was a "major" eruption in Indonoseia in 2004 and one in Colombia in 2008. Other lists include eruptions of Vesuvius in 1906 and 1944 as well as several in Africa since 1800...sigh...I'm not sure what to do. Shall we stick with the main list, plus the two post-2000 eruptions?

Edited December 17, 2008 by Captain_Bobski

[Admiral_Bobski]

Aha! It seems that the definition of "Major" in the list above is "more than 500 fatalities" and nothing to do with the actual size of the eruptions at all. A much smaller list, below, shows the largest eruptions in terms of volumetric output:

Tambora, Indonesia - 1815 (100 cubic km)

Krakatau, Indonesia - 1883 (18 cubic km)

Katmain, Alaska - 1912 (12 cubic km)

Pinatubo, Philippines - 1991 (10 cubic km)

Mt St Helens - 1980 (1 cubic km)

How do you want to handle the numbers - debris volume divided by two as a percentage of light blocked out, just as a ballpark figure which keeps eruption size proportionate?

B)

CB

EDIT - Hold that thought - I'm working on another list based on the Volcanic Explosivity Index.

Sorry to mess you around - I've obviously come down with a bad case of the web forum equivalent of verbal diarrohea!

Edited December 17, 2008 by Captain_Bobski

[Admiral_Bobski]

Okay - I'm back! I'll just do this quickly before everyone gets thoroughly sick of the sight of me!

Based on the Volcanic Explosivity Index, here's every eruption ranked 5 (over 1 cubic km), 6 (over 10 cubic km), and 7 (over 100 cubic km):

Mount Tarumae 1739 (VEI5)

Mount Agung 1963 (VEI5)

Mount St Helens 1980 (VEI5)

El Chichon 1982 (VEI5)

Mount Hudson 1991 (VEI5)

Mount Tarawera 1886 (VEI5)

Laki 1783 (VEI6)

Krakatoa 1883 (VEI6)

Santa Maria 1902 (VEI6)

Novarupta 1912 (VEI6)

Mount Pinatubo 1991 (VEI6)

Mount Tambora 1815 (VEI7)

In date order:

1739 (5)

1783 (6)

1815 (7)

1883 (6)

1886 (5)

1902 (6)

1912 (6)

1963 (5)

1980 (5)

1982 (5)

1991 (5)

1991 (6)

Roughly speaking, a VEI 6 eruption is 10 times bigger than a VEI 5 eruption, and a VEI 7 eruption is 10 times bigger than a VEI 6 eruption (and hence a VEI 7 is 100 times bigger than a VEI 5).

The effect of each eruption should mirror this, so maybe a VEI 5 should block out 10% of solar radiation, a VEI 6 should block 20% and a VEI 7 should block 40%. How does that sound?

B)

CB

[Chris Knight]

Here's an Excel file based on Ice core SO₂ data, showing Northern, Tropical and Southern hemisphere eruption dates, from Marc Duran Prohom's PhD thesis, together with Mann et al's 1998 volcanic forcing data, references on the worksheets.volcanodates1.xls

Some large eruptions, like Mt St Helens, had virtually no climatic effect due to the lateral direction of the ash plume, rather than vertically upwards into the upper atmosphere, and relatively small amounts of sulphate aerosols emitted. Other eruptions like Laki had considerable duration of significant emissions compared to one-off explosive eruptions.